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I am trying to calculate scattering amplitudes with linear sigma model Lagrangian, given as $$\mathcal L= \frac{1}{2}(\partial_{\mu}\sigma)^2+\frac{1}{2}(\partial_{\mu}\vec{\pi})^2-\mathcal U(\sigma,\vec{\pi})\tag{112}$$ Where, $$\mathcal U(\sigma,\vec{\pi})=\frac{\mathcal{\lambda}}{4}(\sigma^2+\vec \pi^2 -\mathcal f^2)^2-{H\sigma}.\tag{113}$$ the scattering amplitudes, for $\pi\pi\rightarrow\pi\pi$ and $\sigma\sigma\rightarrow\sigma\sigma$ are

$$\mathcal M_{{\pi}^a{\pi}^b;{\pi}^c{\pi}^d}=-2\lambda\Bigl(\frac{s-m_\pi^2}{s-m_\sigma^2}\delta_{ab}\delta_{cd}+\frac{t-m_\pi^2}{t-m_\sigma^2}\delta_{ac}\delta_{bd}+\frac{u-m_\pi^2}{u-m_\sigma^2}\delta_{ad}\delta_{bc}\Bigr)\tag{118}$$

$$\mathcal M_{{\sigma}{\sigma};{\sigma}{\sigma}}=-6\lambda-36\lambda^2\mathcal f_\pi^2\Bigl(\frac{1}{s-m_\sigma^2}+\frac{1}{t-m_\sigma^2}+\frac{1}{u-m_\sigma^2}\Bigr)\tag{121}$$

Where $s$,$t$ and $u$ are Mandelstam variables and $$\mathcal f_\pi^2=\frac{m_\sigma^2-m_\pi^2}{m_\sigma^2-3m_\pi^2}\mathcal f^2.\tag{117}$$ I do not know how to start, to get last two expressions (118) & (121), namely, $\mathcal M_{{\pi}^a{\pi}^b;{\pi}^c{\pi}^d}$ and $\mathcal M_{{\sigma}{\sigma};{\sigma}{\sigma}}$.

References:

  1. http://arxiv.org/abs/arXiv:1006.0257
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  • $\begingroup$ The expressions are written in the article arxiv.org/abs/arXiv:1006.0257 and i am following Peskin's book. $\endgroup$ – Suraj Kumar Rai Dec 27 '18 at 19:49
  • $\begingroup$ How to get a one-loop scattering amplitude starting from a Lagrangian is basically the content of most textbooks on QFT, including Peskin's book. A very first step is rewriting your Lagrangian in terms of the vacuum expectation and fluctuation $v$ and $\Delta$ that appear in that paper. $\endgroup$ – octonion Dec 27 '18 at 22:36

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